List of Figures

1. Square Limit by M. C. Escher
2. Square Limit progression
3. Water-wheel imitating the convection system of Lorenz[14].
4. The Koch Snowflake
5. The asymptotically self-similar Hilbert curve. Notice the effect of aliasing when the details exceed the resolution of the printer or screen.
6. A simple Weierstrass function with and $k = 5,6,...,12$. (a) shows the starting 346 samples at sampling rate of 22050 Hz and (b) shows exactly half of that signal (the starting 173 samples). Notice that the two shapes differ only in high frequency details. This similarity can be seen in higher or lower time scales as well.
7. A Weierstrass function with $\beta = 2.1189$ and $k = 5,6,...,12$. (a) and (b) are the first 205 and 97 ( $205 / 2.1189$ ) samples.
8. The time domain and power spectrum of white noise with average duration of 0.1 seconds. (a) illustrates the first 30 seconds of the signal and (b) is the power spectrum of such a signal in log-log scale. The line representing the $1/f$ is also drawn for reference. Notice that the power spectrum for this signal is flat for the area of our inspection which is between 0.001 and 5 Hz.
9. The time domain and (log-log scale) power spectrum of random signals with average duration of 0.5, 2, and 200 are illustrated. The line representing the $1/f$ line is also drawn for reference. Notice that the power spectrums show a slope steeper than the $1/f$ line in the area of correlation while the rest of the spectrum stays flat.
10. In this figure the effect of bad quantization of a random signal is illustrated. The values of a random signal are truncated to the quantized level rather than being rounded to the nearest level. This figures should be compared to the first illustration of figure 4-4. Notice how the low frequency power increases as we use fewer quantization levels.
11. The first 30 seconds and the power spectrum of a random signal with a simple deterministic shape added to it is illustrated. The deterministic shaped is scaled to the duration of the note. Notice that such a process shows up as low frequency (i.e. long term correlation) on the power spectrum.
12. (a) is the first 30 seconds of the ``top voice'' signal extracted from J. S. Bach's 3rd Brandenburg Concerto. (b) is the the power spectrum of the ``top voice'' signal for the duration of the piece. Notice that the line representing the $1/f$ line fits the slope of the power spectrum.
13. The spectrum of two of the odd cases of the analysis is shown. Figure (a) is the power spectrum of the ``top voice'' signal of the 8th prelude from J. S. Bach's Well-tempered Clavier Part I. Notice that the slope of the spectrum is sharper than the $1/f$ line and that can be explained by the static melody of the top voice in that piece. Figure (b) is the power spectrum of the 11th prelude from J. S. Bach's Well-tempered Clavier Part I. Notice that the spectrum is flat in the 0.1-5 Hz region. This effect was caused by the way we extracted the top voice. The interaction between the half note trills and our down-sampling of the MIDI data created a noisy melody which is characterized by a flat spectrum.
14. The frequency fluctuation of the audio example 4 is illustrated. (a) shows the frequency fluctuation in 2 seconds and (b) shows the frequency fluctuation in 1 second. The basic shape of both graphs are similar to each other.
15. The frequency fluctuation of the audio example 5 is illustrated. (a) shows the frequency fluctuation in 2 seconds and (b) shows the frequency fluctuation in 0.667 second. The basic shape of both graphs is a triangle.
16. The spectrogram of the first 60 and the first 3 seconds of audio example 8 is illustrated. The spectrogram of the first 3 seconds is rescaled by a factor of 0.4. As it can be seen, the same structure is manifested in this sound in two levels of our auditory perception.
17. The spectrogram of audio example 10 is illustrated.
18. The spectrogram of sound example 11 is illustrated.
19. The time segmentation of the first 4 levels for audio example 15 is illustrated. This example has a two-level hierarchy. The first level goes through a binary segmentation, and the first part of the second level goes through a trinary segmentation. The time segments which have gone through a trinary segmentation are shaded.